Primordial magnetic fields from cosmological phase transitions

We study the generation of large scale primordial magnetic fields by a cosmological phase transition during the radiation dominated era. The setting is a theory of N charged scalar fields coupled to an abelian gauge field, that undergoes a phase transition at a critical temperature much larger than the electroweak scale. The dynamics after the transition features two distinct stages: a spinodal regime dominated by linear long-wavelength instabilities, and a scaling stage in which the non-linearities and backreaction of the scalar fields are dominant.This second stage describes the growth of horizon sized domains.We implement a formulation based on the non-equilibrium Schwinger-Dyson equations to obtain the spectrum of magnetic fields that includes the dissipative effects of the plasma. We find that large scale magnetogenesis is efficient during the scaling regime.Charged scalar field fluctuations with wavelengths of the order of the Hubble radius induce large scale magnetogenesis via loop effects. The leading processes are: pair production, pair annihilation and low energy bremsstrahlung, these processes while forbidden in equilibrium are allowed strongly out of equilibrium.The ratio between the energy density on scales larger than L and that in the background radiation r(L,T)= rho_B(L,T)/ rho_{cmb}(T) is r(L,T) sim 10^{-34} at the Electroweak scale and r(L,T) sim 10^{-14} at the QCD scale for L sim 1 Mpc.The resulting spectrum is insensitive to the magnetic diffusion length and equipartition between electric and magnetic fields does not hold. We conjecture that a similar mechanism could be operative after the QCD chiral phase transition.